Problem: $ A = \left[\begin{array}{rr}1 & 2 \\ 3 & 4 \\ -2 & 3\end{array}\right]$ $ F = \left[\begin{array}{rr}3 & -2 \\ 3 & 0\end{array}\right]$ What is $ A F$ ?
Answer: Because $ A$ has dimensions $(3\times2)$ and $ F$ has dimensions $(2\times2)$ , the answer matrix will have dimensions $(3\times2)$ $ A F = \left[\begin{array}{rr}{1} & {2} \\ {3} & {4} \\ \color{gray}{-2} & \color{gray}{3}\end{array}\right] \left[\begin{array}{rr}{3} & \color{#DF0030}{-2} \\ {3} & \color{#DF0030}{0}\end{array}\right] = \left[\begin{array}{rr}? & ? \\ ? & ? \\ ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ A$ , with the corresponding elements in column $j$ of the second matrix, $ F$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ A$ with the first element in ${\text{column }1}$ of $ F$ , then multiply the second element in ${\text{row }1}$ of $ A$ with the second element in ${\text{column }1}$ of $ F$ , and so on. Add the products together. $ \left[\begin{array}{rr}{1}\cdot{3}+{2}\cdot{3} & ? \\ ? & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ A$ with the corresponding elements in ${\text{column }1}$ of $ F$ and add the products together. $ \left[\begin{array}{rr}{1}\cdot{3}+{2}\cdot{3} & ? \\ {3}\cdot{3}+{4}\cdot{3} & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ A$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ F$ and add the products together. $ \left[\begin{array}{rr}{1}\cdot{3}+{2}\cdot{3} & {1}\cdot\color{#DF0030}{-2}+{2}\cdot\color{#DF0030}{0} \\ {3}\cdot{3}+{4}\cdot{3} & ? \\ ? & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rr}{1}\cdot{3}+{2}\cdot{3} & {1}\cdot\color{#DF0030}{-2}+{2}\cdot\color{#DF0030}{0} \\ {3}\cdot{3}+{4}\cdot{3} & {3}\cdot\color{#DF0030}{-2}+{4}\cdot\color{#DF0030}{0} \\ \color{gray}{-2}\cdot{3}+\color{gray}{3}\cdot{3} & \color{gray}{-2}\cdot\color{#DF0030}{-2}+\color{gray}{3}\cdot\color{#DF0030}{0}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rr}9 & -2 \\ 21 & -6 \\ 3 & 4\end{array}\right] $